Geremia and Sullivan [Ann. Math Pure Appl. 127 (1981), 231–251] gave a necessary and sufficient condition for an ℓ^p-product of spaces to be 2-uniformly rotund. We extend this result to k-uniform rotundity for any integer k > 1. The nonreflexive, uniformly nonoctahedral Banach space \(\widetilde{X}\) constructed by James [Israel J. Math. 18 (1974), 145–155] does not contain arbitrarily close copies of although it is not k-uniformly rotund (k-UR) for any k ⩾ 2. This shows that a Banach space X not being k-UR does not imply that X contains arbitrarily close copies of ℓ^k+1₁ for each k ⩾ 2. We show that a sufficient condition that a Banach space X be not k-UR is that it contains an arbitrarily close copy of one of the faces of ℓ^k+1₁ rather than itself.

中文翻译：

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Banach空间中凸性和圆度的多维模

Geremia和Sullivan [ Ann。数学纯应用程序。 127（1981），231-251]，获得必要且充分条件的ℓ ^p的空间-产物为2-一致凸点。对于任何大于k的整数k，我们将结果扩展到k-均匀圆度。由James [ Israel J. Math。Chem。，2000，pp。177]构造的非自反，一致非八面体的Banach空间\（\ widetilde {X} \）。18（1974），145-155]中不包含的虽然没有任意接近拷贝ķ -uniformly圆嘟嘟（ķ -ur）对于任何ķ ⩾2.由此可见，Banach空间X不是ķ -ur并不意味着X包含的任意接近拷贝ℓ ^{ķ 1} ₁ 对于每个ķ ⩾2.我们表明，一个充分条件Banach空间X来不ķ -ur是它含有的一个的任意接近拷贝的面ℓ ^{ķ 1} ₁ ，而不是本身。